Thick Lens

Thick Lens 0Thick Lens

Image formation by a water droplet.


A thick lens comprises a system of two refracting surfaces, 1 and 2 , separated by an axial distance t (the lens thickness) where t is not a negligible dimension.

Thick Lens 2Thick Lens in a Uniform Medium

We can apply the General Thick Lens Equation but it’s cumbersome.

with:


Consider incident parallel (to OA) light rays on a thick lens. These rays will converge at the 2nd focus, F’ , of the lens.

OA


We can define a plane by the collection of intersection points obtained upon extrapolating the incident parallel rays forward and the converging rays (from F’ ) backward.

This is the “2nd Principal Plane” : H’

Intersection of the optic axis (OA) with H’ gives the “2nd Principal Point” : H’

Thick Lens 4aThick Lens

2nd Principal Plane : H’

1 and 2 are the bounding surfaces of the lens.


Consider parallel light rays emerging from a thick lens. Such rays originate at the 1st focus, F , of the lens.

OA


We can define a plane by the collection of intersection points obtained upon extrapolating the emerging parallel rays backward and the incident rays (from F ) forward.

This is the “1st Principal Plane” : H

Intersection of the optic axis (OA) with H gives the “1st Principal Point” : H

Thick Lens 6aThick Lens

1 and 2 are the bounding surfaces of the lens.

1st Principal Plane : H


Having identified principal planes / points, we now define:

the 2nd focal length of the thick lens.

the 1st focal length of the thick lens.

We can prove (later) that, for a thick lens in a uniform medium:


Thus, we have:

Thus, the identifications FH f and F’H’ f ’ give the Newtonian Thin Lens Equation.


Thus, we have:

Also, the transverse magnification is given by:

Thick Lens 10Thick Lens Next, we define image / object distances with respect to the principal planes.

By the Newtonian Thin Lens Equation:

uniform medium

Thick Lens 11Thick Lens Thus, by the Newtonian Thin Lens Equation:

Divide by ss’f to get:

Thin Lens Equation (uniform medium)

Thick Lens 12Thick Lens Also:

as for the Thin Lens

Thick Lens 13Thick Lens New Sign Convention:

Note: s’ < 0 no longer signifies virtual image.

s’ > 0 no longer signifies real image.

Thick Lens 14Thick Lens Example:

s’ < 0 and real image. s > 0 and virtual object.

Principal Planes 1Principal Planes: Generalization

For any optical system (no matter how complex) we can define principal planes by ray tracing:

Principal Planes 2Principal Planes: GeneralizationF , F’ , H , H’ are the Cardinal Points of the Optical System

Once we locate the Cardinal Points of an optical system we can draw simple ray diagrams and use all Thin Lens results!

Principal Planes 3Principal Planes: GeneralizationF , F’ , H , H’ are the Cardinal Points of the Optical System

The points V and V’ (vertices) define the first and last refracting surface locations of the system (ie. the bounding surfaces). Planes H and/or H’ can be inside or outside of the bounding surfaces.


With s , s’ , f , f’ referenced to H , H’ the Thin Lens results hold.

Uniform medium


For a thin lens, the principal planes H and H’ are both located at the lens midplane.

Principal Planes 6Principal Planes: GeneralizationFor system in non-uniform medium, general thin lens results hold.


Note: the principal planes H and H’ are not necessarily within the physical confines of the system.

For example

V and V’ are vertices of the bounding surfaces of the system.

Sometimes H and H’ are “crossed” H closer to F’ than H’and H’ closer to F than H. Otherwise they’re “normal”.


Surfaces and ’ are bounding surfaces of the system (with vertices V and V’). Distances h and h’ locate H and H’ with respect to V and V’ respectively.

Sign convention:h > 0 for H right of V h’ > 0 for H’ left of V’

Cardinal points thick lens 1Cardinal Points of Thick LensSet an object point at infinity (on OA) to find F’

Cardinal points thick lens 2Cardinal Points of Thick LensSet an object point at infinity (on OA) to find F’ .

Apply Gauss’ Formula at the two surfaces:

This gives s2’ and locates F’ with respect to V2 .

Cardinal points thick lens 3Cardinal Points of Thick Lens

Find the focal length f = f ’ for a uniform medium.

Cardinal points thick lens 4Cardinal Points of Thick Lens

Cardinal points thick lens 5Cardinal Points of Thick LensWe have located H’ , F’ with respect to V2

With s2’ given by:

With f given by:

“Turn lens around” to find H , F with respect to V1

Cardinal points thick lens eg 1Cardinal Points of Thick Lens: ExampleFind the cardinal points of the following lens.

Radius of curvature of 1 : R1 = 100 mm (> 0) Radius of curvature of 2 : R2 = 200 mm (> 0) Axial thickness : t = 10mm n = 1.50

Cardinal points thick lens eg 2Cardinal Points of Thick Lens: ExampleSolution: (1) Find F’ and H’ with respect to V2

Find s2’ and f .

Cardinal points thick lens eg 3Cardinal Points of Thick Lens: Example

Find s2’ and f .



Solution: (2) Find F and H with respect to V1 . Turn lens around and repeat exercise.


Find s2’ and f .


Cardinal points thick lens eg 8Cardinal Points of Thick Lens: ExampleThus, for our thick lens, the cardinal points are located as follows:

Focal length thick lens 1Focal Length of Thick LensWe have a system of 2 refracting surfaces:

We have the expression for the thick lens focal length, eqn (1):

Focal length f1’ is focal length of refracting surface 1 , eqn (2) :

Focal length thick lens 2Focal Length of Thick LensFor an object point at infinity, the final image lies at the 2nd focus of the system. To find it we apply Gauss’ formula at 2 :

Eqn (3) :

Combine equations (1) , (2) and (3) :

Multiply by:

Focal length thick lens 3Focal Length of Thick LensWe get:

Using equations (1) and (2):

Rewriting:

Focal length of Thick Lens in Air

(measured with respect to principal planes)

Cardinal points 2 lens sys 1Cardinal Points of Two Lens System

As for thick lens, take an object point at infinity. Intermediate image due to lens1 alone lies at F1’ a distance f1’ = f1 from lens1. Use thick lens result:


F’ is 2nd focus of system and F1’ is 2nd focus of Lens1.


Intermediate image at F1’ acts as object for lens2. Distance s2is object distance for lens2. Distance s2’ is image distance for lens2. This image is the final image of the system for an object point at infinity. Thus s2’ locates F’ of the system with respect to vertex V2. Distance s2’ is obtained by using TLE at lens2.


Combining results:


Rewriting:

Formula for the focal length of a two lens system (measured with respect to principal planes.


Find the principal points H and H’


We have:

We require:


Result:

Turn lens system around (interchange lens1 and lens2) and do similar calculation to find h.

(Details on next page)

h’ > 0 H’ left of L2

h > 0 H right of L1


Details:


Note: for thin lenses in contact d =0 :

Note: for N thin lenses in contact:

Sometimes written as:

Where P is “refractive power”.

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